Two-phase Pipe Flow Research Articles

Experimental study on vertical downward air-water two-phase flow in a large diameter pipe

Downward two-phase flows in large diameter pipes are important in various industrial applications, especially for the safety analysis in nuclear reactors. To address the issue that few data of downward flow in large diameter pipes is available for model evaluation, experiments of air–water downward flow in a pipe with inner diameter of 203.2 mm have been performed. Area-averaged void fraction and pressure measurement, as well as flow visualization, have been conducted at several axial locations. The flow conditions for superficial gas velocity range from 0.05 m/s to 3.00 m/s and for superficial liquid velocity range from 0.1 m/s to 1.5 m/s, which cover cap-bubbly flow, churn-turbulent flow and annular/falling film flow. The flow structure at several axial locations and the transition from churn-turbulent flow to annular/falling film flow have been discussed. Current available drift-flux models developed for downward flow in regular pipes as well as for upward flow in large pipes are evaluated using newly collected data. For churn-turbulent flow, the data indicates a larger drift velocity than the model prediction. Corresponding drift-flux constitutive equations are suggested which can reduce the prediction error from 34.37% to 11.79%.

Liquid droplet entrainment in two-phase oil-gas low-liquid-loading flow in horizontal pipes at high pressure

An experimental study is conducted in a 152.4-mm inner diameter (ID) high-pressure facility to investigate the liquid droplet entrainment in two-phase oil-gas flow in horizontal pipe under low-liquid loading condition. Total of 247 tests are carried out at five different operating pressures, namely, 1.48, 1.82, 2.17, 2.51, and 2.86 MPa. The test fluids are nitrogen and mineral oil (Isopar-L). Superficial oil velocity (vSL) ranges from 0.01 to 0.05 m/s, while superficial gas velocity (vSg) ranges from 1.6 to 16.7 m/s depending on the operating pressures. An isokinetic sampling system is used to measure the entrained droplet flux at three different vertical positions in the gas phase. A high-speed video camera is used to visually observe and estimate the onsets of entrainment. This paper provides a unique entrainment data set for two-phase flow in a large diameter pipe at high operating pressure.The experimental data are analyzed to evaluate the effects of high operating pressure and superficial velocities. It is found that the entrainment fraction increases with the operating pressure. In addition, predictions of ten correlations for two-phase entrainment fraction are compared with the experimental data to evaluate the performances. The results show that these correlations over-predict the entrainment fraction data from this study.

The effect of gas flow rate on the wax deposition in oil-gas stratified pipe flow

Wax deposit becomes more complicated in oil-gas two-phase pipe flow than single phase flow, and therefore remains poorly understood. The increasing trend of deposit with an increasing gas flow rate has been observed in wax deposit experiments under oil-gas stratified pipe flow. In this study, a numerical method is used to clarify this trend by quantitatively analyzing the heat and mass transfer during wax deposit building up. The method was able to predict the experimentally observed trend of deposit without any adjustable parameter. It was found that there are four effects to affect wax deposit when the gas flow rate changed. These four effects focus on the flow, heat and mass transfer characteristic at the oil-deposit interface, which include the effect of oil wetted area, the wax diffusivity, the boundary layer thickness, and the concentration difference between bulk oil and oil-deposit interface on mass transfer. Furthermore, the overall growth behavior of the wax deposit is the consequence of the competition between these four effects as time progresses. These results provided important insight about the effects of the gas flow rate on the wax deposit in oil-gas stratified pipe flow.

CFD study of oil-in-water two-phase flow in horizontal and vertical pipes

The effectiveness of the classical Lagrangian method implemented in 3D CFD solver to simulate highly turbulent vertical and horizontal liquid-liquid dispersed flows where intensive turbulent migration of droplets make the flow behaviour complex is in the focus of the study. Dispersed oil-in-water two-phase pipe flow is analysed numerically using CFD solver ANSYS Fluent where a three-dimensional mathematical model of two-phase flow in pipes is developed. The Euler-Lagrange scheme is employed to resolve the interaction between the fluid (water) and droplet (oil) phases. The flow field of the continuous fluid phase is calculated by solving the Reynolds-averaged Navier-Stokes (RANS) conservation equations that are coupled with a high Reynolds number k-ε turbulence model and standard wall function. The Lagrangian method is used to resolve the motion of oil droplets. The drag, gravity, buoyancy and shear-lift forces are considered. The shear-lift force is of primary focus and is modelled and incorporated into the governing equations of the solver through the User Define Function (UDF) option as the external program. The two-way coupling procedure is employed. Here, droplet breakup and coalescence are not modelled. Mean droplet size is calculated using analytical methods. The model is validated with high level of accuracy against published experiments where the frictional pressure drop in vertical and horizontal dispersed oil-water flows is measured. The effect of mixture flow velocity and fluid phase superficial velocity on the droplet dynamics in vertical and horizontal pipes is investigated numerically. The transition from dispersed two-phase flow to stratified flow regime is analysed for the case of horizontal pipe flow. It has been found that the shear-lift force has much higher importance for the adequate flow representation as compared to the break-up/coalescence phenomena in the case of highly turbulent liquid-liquid dispersed pipe flows.

Application of a single wire-mesh sensor in a parabolic trough facility with direct steam generation

The direct steam generation process can contribute to increase the cost-effectiveness of line-focused solar thermal power plants. The current research focuses on the once-through mode as very promising option for large-scale commercial plants. However, the existing pool of knowledge about the steam–water flow in the horizontal receiver pipes reveals knowledge gaps. The present contribution aims to experimentally investigate the two-phase flow in the receiver pipes of the DISS test facility at the Plataforma Solar de Almería, Spain, by an invasive measurement device for the first time. Therefore, a conductance wire-mesh sensor is employed in the low-quality evaporation section of the DISS test facility. This measurement device measures the local instantaneous void fraction that is used to describe the prevailing flow pattern under different pressures and mass flow rates. The results show slug flow, roll waves, wavy flow, and stratified-wavy flow in the horizontal receiver pipes within pressures of 30–80bar. The probability of slug flow or large-amplitude waves decreases with higher pressures. The findings provide a valuable knowledge about the flow structure within the receiver pipes and confirm the proof of function of the wire-mesh sensor under conditions of the direct steam generation process.

Ultrasonic Measurement of Velocity Profile on Bubbly Flow Using Fast Fourier Transform (FFT) Technique

In two-phase bubbly flow, measurement of liquid and bubble velocity is a necessity to understand fluid characteristic. The conventional ultrasonic velocity profiler (UVP), which has been known as a nonintrusive measurement technique, can measure velocity profile of liquid and bubble simultaneously by applying a separation technique for both phases (liquid and bubble) and transparent test section is unnecessary. The aim of this study was to develop a new technique for separating liquid and bubble velocity data in UVP method to measure liquid and bubble velocity profiles separately. The technique employs only single resonant frequency transducer and a simple UVP system. An extra equipment is not required. Fast Fourier Transform (FFT) based frequency estimator paralleled with other signal processing techniques, which is called as proposed technique, was proposed to measure liquid and bubble velocity separately. The experimental facility of two-phase bubbly flow in the vertical pipe was constructed. Firstly, the Doppler frequency estimation by using the FFT technique was evaluated in single-phase liquid flow. Results showed that FFT technique showed a good agreement with autocorrelation and maximum likelihood estimator. Then, separation of liquid and bubble velocity was demonstrated experimentally in the two-phase bubbly flow. The proposed technique confirmed that liquid and bubble velocity could be measured efficiently.

Influence of boiler size and location on one-dimensional two-phase vertical pipe flow

We study the evolution of flow and temperature of a fluid moving upstream in a long, thin vertical pipe when a boiler element is involved. The main goal of this work is to understand how the size and position of the boiler will affect the flow and temperature in the pipe over time, as current literature considers cases where the heater or boiler covers the whole length of the pipe, or when already boiling fluid enters a pipe without a boiler. Therefore, we shall allow for a boiling element which covers only a fraction of the pipe when devising out mathematical model. The boiling process results in a transition to different multiphase flow regimes, and we therefore consider a two-phase flow model. From this model, we obtained a simplified one-dimensional model, since we are concerned with a long, thin pipe, under reasonable assumptions and reductions which still preserve the desired physics. We performed a stability analysis for the boiling boundary denoting the phase change in this model. We then obtained numerical simulations for the steady and transient solutions. The numerical results suggest that both the size and position of the boiler strongly affect the flow regime. In particular, depending on the size of the boiler, transition to other phases might not always occur, and depending on its position along the pipe, the fluid coming out at the top of the pipe might not have the desired thermal profile. As such, one may tailor the position and size of the boiler element in order to obtain a useful thermal profile for particular applications. Such results are of possible relevance in industrial applications where heating or boiling of fluid is required.

A Study of Flow-Pattern Transitions in High-Viscosity Oil-and-Gas Two-Phase Flow in Horizontal Pipes

SummaryTwo-phase flow pattern is an important parameter for predicting pressure gradient and liquid holdup in mechanistic models, both of which are required to design and operate hydrocarbon production and transportation systems. Existing two-phase-flow mechanistic models have shown various degrees of discrepancy in predicting flow pattern and their transitions for immiscible laboratory-scale flows of gas and high-viscosity Newtonian liquids (mineral oil) caused by the additional complexity that a high liquid viscosity introduces to the two-phase flow behavior. The objective of this study is to improve the high-viscosity oil flow-pattern transition modeling in horizontal pipes for the stratified-smooth (SS)/stratified-wavy (SW) and intermittent/annular transitions. These two transitions were improved by introducing a liquid viscosity-dependent sheltering coefficient model, and a new high-viscosity liquid-level criterion, respectively. An additional objective is to investigate the existing inviscid Kelvin-Helmholtz (IKH), viscous Kelvin-Helmholtz (VKH), and Taitel and Dukler models for the stratified/nonstratified (S–NS) transition, and provide insights on their applications for high-viscosity oil-flow patterns. A validation study of the proposed transition models is carried out with four laboratory-scale two-phase gas/oil flow-pattern experimental data sets covering a wide range of oil viscosities. The validation study revealed that the proposed SS/SW, and intermittent/annular transition models predicted the considered high-viscosity, horizontal flow-pattern data with more accuracy than the Taitel and Dukler and Barnea VKH S/NS models. Furthermore, the validation and comparison of IKH, VKH, and Taitel and Dukler S/NS models show that the IKH model is the simplest one that sufficiently predicts stratified flow for high-viscosity oil, especially for liquid viscosity above 100 mPa·s.

Twin Rectangular Fork-Like Capacitance Sensor to Flow Regime Identification in Horizontal Co-Current Gas–Liquid Two-Phase Flow

In this paper, a twin rectangular fork-like capacitance (TRFLC) sensor to recognize the flow regime of gas-oil two-phase flow in a horizontal pipe is proposed. Two phase flow experiment is carried out in a 26.6-mm ID horizontal pipe to determine the response of TRFLC sensor on different flow regimes. Both measuring and exciting electrodes consist of a base and double rectangle blades, which are placed alternately. The finite-element method (FEM) is used to calculate the sensitivity field of the sensor. The effect of sensor geometry on the distribution of sensitivity field and the optimal sensor structure is presented. Based on FEM and experimental results, the sensitivity of TRFLC was higher than the previous concave electrode (overall sensitivity of TRFLC 3.27 pF and concave 2.19 pF). Simulation results show that two blades for each electrode have more sensitivity than three or four blades in the regions near to the center of the pipeline and more uniform sensitivity in whole section of the pipe. The flow variations especially in the center of the pipeline were traced with higher sensitivity than concave type. As a result, the flow regimes such as slug can be identified with more accuracy. The flow regimes were classified using wavelet transforms and power spectral density (PSD). The wavelet variance is not sufficient to flow regime assignment due to some flow regimes overlapping. The wavelet variance in conjunction with the cumulative PSD solves this problem and presents a very useful tool to flow regime classification.

A dual grid method for the compressible two-fluid model which combines robust flux splitting methodology with high-resolution capturing of incompressible dynamics

The speed of sound in two-phase pipe flow systems is often several orders of magnitude greater than the travelling speed of hydraulic information (volume fractions). Dynamically simulating such flows requires resolution of acoustic and hydraulic waves existing at vastly different spatial and temporal scales. If simulated on the same numerical grid, the need for accuracy in hydraulic waves will necessitate an exaggerated resolution of acoustic waves. Likewise, time steps restricted by the speed of sound are small compared to the time scales active in hydraulic waves. This constitutes a waste of computational potential. The method proposed herein decouples the hydraulic and acoustic scales, greatly improving computational efficiency.The proposed dual grid method solves a four-equation compressible two-fluid model on a principal grid which robustly accounts for the pressure evolution and conserves mass and momentum. An incompressible two-equation model is at the same time solved on a finer grid, resolving the details of the hydraulic evolution. Information from both model formulations is coupled through the terms of the governing transport equations, providing consistency between the grids. Accurate and finely resolved schemes can then be employed for the incompressible two-fluid model without suffering from the time and stability restrictions otherwise enforced by acoustic waves. At the same time, the Hybrid Central-Upwind flux splitting scheme of Evje and Flåtten (2005a) allows for an explicit and numerically robust treatment of the acoustic waves without losing hydraulic accuracy.The dual grid method is tested against four dissimilar problems: A shock tube problem, the water faucet problem, a surge wave and pressure wave problem and a roll-wave case. In all problems, the proposed scheme provided significant increases in computational efficiency and accuracy as compared with a single grid arrangement.

Two-phase flow pipe network method for simulation of CO2 sequestration in fractured saline aquifers

A two-phase flow unified pipe network method is developed to simulate CO2 evolution in fractured saline aquifers. Fractures are explicitly represented in the proposed method. The two-phase flow in both rock matrix and fractures is considered by using different equivalent pipe flow models respectively, namely the two-phase matrix pipe flow model and the two-phase fracture pipe flow model. The equivalent flow coefficients of the pipe flow models are derived based on flow rate equivalence. The coupling of the fracture pipe flow and matrix pipe flow is treated by applying the extended capillary pressure conditions. Brooks-Corey relative permeability model and capillary model are adopted to simulate CO2 (non-wetting phase) invasion into the brine (wetting phase) saturated formation, which is a typical drainage process. Accurate Equations of State for calculating density and viscosity of CO2 are incorporated to reflect its change in hydraulic characteristics during the injection processes. The proposed method is simple yet robust and not sensitive to the mesh quality. The complex geological and topological features of fracture networks can, therefore, be well retained in the proposed method. The anisotropy and heterogeneity characteristics of the fractured rock mass caused by the fracture networks can be accurately represented. The proposed method is verified by comparing to other numerical methods. Both 2D and 3D models with complex fracture networks are presented to demonstrate the feasibility of proposed method. Numerical examples show that fractures can significantly affect the distribution and evolution of CO2 in aquifers and the differences of entry capillary pressures for fractures and matrix rock should be accurately simulated.

Experimental investigation of two-phase pipe flow with ultrafast X-ray tomography and comparison with state-of-the-art CFD simulations

Abstract In this paper, we report on experimental investigations of air/water two-phase flow in a vertical pipe, which is a benchmark experiment for the development and validation of two-phase CFD models and codes. We conducted experiments for co-current upward and downward flow as well as counter-current flow at ambient temperature and pressure conditions. As measurement technique, we employed ultrafast X-ray tomography, which gives two-dimensional cross-sectional images of the gas-liquid distribution in two planes up to 2500 Hz scanning speed. In this paper, we briefly explain this technique, introduce essential data processing steps and discuss the findings of the study.

Modeling of void fraction covariance and relative velocity covariance for upward boiling flow in vertical pipe

Drift-flux parameters have been often used to formulate one-dimensional interfacial drag force in dispersed two-phase flow, which is one of key parameters to predict void fraction using one-dimensional thermal-hydraulic codes. This approach is called “Andersen approach”, which has been widely used in one-dimensional nuclear thermal-hydraulic system analysis codes such as TRACE, RELAP5 and TRAC-BF1. However, the current formulation of one-dimensional interfacial drag force ignores important void fraction covariance and relative velocity covariance when local interfacial drag force is converted to one-dimensional interfacial drag force. The impact of neglecting void fraction covariance and relative velocity covariance on one-dimensional interfacial drag force and relative velocity has been discussed in detail. In view of the importance of the drift-flux parameters, void fraction covariance and relative velocity covariance on one-dimensional formulation of the interfacial drag force, three constitutive equations have been developed for upward boiling two-phase flow in a vertical pipe. The validity of the modeled void fraction covariance and relative velocity covariance for subcooled and bulk boiling flow in a vertical pipe has been verified by boiling R12 data taken in a vertical pipe with the diameter of 19.2mm under the pressure simulating prototypic nuclear reactor thermal-hydraulic conditions. The correlation of void fraction covariance agrees with the boiling flow data in the vertical pipe with the mean absolute error, standard deviation, mean relative deviation and mean absolute relative deviation being 0.828, 3.43, 10.3% and 33.5%, respectively. The correlation of relative velocity covariance agrees with the boiling flow data in the vertical pipe with the mean absolute error, standard deviation, mean relative deviation and mean absolute relative deviation being −0.00394, 0.0663, −0.184% and 5.11%, respectively. Due to the great importance of the void fraction covariance and relative velocity covariance on one-dimensional interfacial drag force formulation, it is highly recommended to include the void fraction covariance and relative velocity covariance in the one-dimensional formulation of the interfacial drag force used in nuclear thermal-hydraulic system analysis codes.

A Computation Fluid Dynamic Model for Gas Lift Process Simulation in a Vertical Oil Well

Abstract Continuous gas-lift in a typical oil well was simulated using computational fluid dynamic (CFD) technique. A multi fluid model based on the momentum transfer between liquid and gas bubbles was employed to simulate two-phase flow in a vertical pipe. The accuracy of the model was investigated through comparison of numerical predictions with experimental data. The model then was used to study the dynamic behaviour of the two-phase flow around injection point in details. The predictions by the model were compared with other empirical correlations, as well. To obtain an optimum condition of gas-lift, the influence of the effective parameters including the quantity of injected gas, tubing diameter and bubble size distribution were investigated. The results revealed that increasing tubing diameter, the injected gas rate and decreasing bubble diameter improve gas-lift performance.

Experimental investigations of the circumferential liquid film distribution of air-water annular two-phase flow in a horizontal pipe

The circumferential film thickness distribution in an air-water horizontal annular two-phase flow was investigated in 26mm pipe diameter by using the conductance probe. The superficial gas and liquid velocities were 10–40m/s and 0.025–0.4m/s, respectively.In general, the film thickness decreases as the liquid velocity decreases and the gas velocity increases. For low superficial gas velocity, high asymmetry of liquid film thickness distribution is generally found. In addition, the liquid film at the upper half of the pipe mostly consists of surge of waves with frequency lower than that of the bottom. At the lowest superficial liquid velocity of the present experimental range, the mean film thickness at the bottom ranges from 0.24mm to 1.3mm. Meanwhile, at the highest superficial liquid velocity, it ranges from 1.0 to 4.29mm, depending on the superficial gas velocity. At the higher superficial gas velocity, a more uniform of liquid film thickness distribution is commonly found. Under this flow condition, the wave at the upper has a similar frequency to that of the lower half and a coherent liquid wave is normally found.To predict the film thickness at any circumferential position, an empirical correlation is proposed on the basis of the physics of the phenomena. In comparison to the available experimental data, the correlation has a good capability to predict the circumferential liquid film thickness distribution.

The nonlinear analysis of horizontal oil-water two-phase flow in a small diameter pipe

Horizontal oil-water two-phase flows are frequently encountered in many industrial processes but the understanding of the dynamic behavior underlying the different flow patterns is still a challenge. In this study, we first conduct experiments of horizontal oil-water flows in a small diameter pipe, and collect the fluctuation signals from conductance probes. The multi-scale power-law correlations of the oil-water flow structures are investigated using detrended fluctuation analysis (DFA) based on the magnitude and sign decomposition of the raw signals. The analysis reveals the scaling behavior of different flow structures; five conductive flow patterns are indentified based on the magnitude and sign scaling exponents at different time scales. In addition, the transfer entropy (TE) in a state space is used to study the information transferring characteristics of the oil-water mixture flowing past a conductance cross-correlation velocity probe. The results of TE indicate that the transferring information depends on the flow conditions and can be used to show changes in the flow patterns.

Interfacial friction in upward annular gas–liquid two-phase flow in pipes

Accurate prediction of interfacial friction between the gas and liquid in annular two-phase flow in pipes is essential for the proper modelling of pressure drop and heat transfer coefficient in pipeline systems. Many empirical relationships have been obtained over the last half century. However, they are restricted to limited superficial liquid and gas velocity ranges, essentially apply to atmospheric pressures, and the relationships are only relevant for pipes with inner diameters between 10 and 50mm. In this study, we carried out experiments in a large diameter flow loop of 101.6mm internal diameter with the superficial gas and liquid ranges of 11–29m/s and 0.1–1.0m/s respectively. An examination of published interfacial friction factor correlations was carried out using a diverse database which was collected from the open literature for vertical annular flow. The database includes measurements in pipes of 16–127mm inner diameter for the liquid film thickness, interfacial shear stress, and pressure gradient for air-water, air-water/glycerol, and argon-water flows. Eleven studies are represented with experimental pressures of up to 6bar. Significant discrepancies were found between many of the published correlations and the large pipe data, primarily in the thick film region at low interfacial shear stress. A correlation for the interfacial friction factor was hence derived using the extensive database. The correlation includes dimensionless numbers for the effect of the diameter across pipe scales to be better represented and better fit the wide range of experimental conditions, fluid properties, and operating pressures.

Gamma-ray CT from incomplete projections for two-phase pipe flow.

A low-energy low-dose γ-ray computed tomography (CT) system used in the gas-liquid two-phase pipe flow measurement has been studied at Tianjin University in recent years. The γ-ray CT system, having a third-generation X-ray CT scanning configuration, is comprised of one 300mCi 241Am source and 17 CdZnTe detector units and achieves a spatial image resolution of about 7 mm. It is primarily intended to measure the two-phase pipe flow and provide improvement suggestions for industrial CT system. Recently we improve the design for image reconstruction from incomplete projection to optimize the scanning parameters and reduce the radiation dose. First, tomographic problem from incomplete projections is briefly described. Next, a system structure and a hardware circuit design are listed and explained, especially on time parameter setting of the pulse shaper. And then a detailed system analysis is provided in Section II, mainly focusing on spatial resolution, temporal resolution, system noise, and imaging algorithm. Finally, we carry on necessary static and dynamic experiments in a full scan (360°) and two sets of partial scan reconstruction tests to determine the feasibility of this γ-ray CT system for reconstructing the images from insufficient projections. And based on an A-variable algebraic reconstruction technique method, a specially designed algorithm, we evaluate the system performance and noise level of this CT system working quantitatively and qualitatively. Results of dynamic test indicate that the acceptable results can be acquired using a multi-source γ-ray CT system with the same parameters when the flow rate is less than 0.04 m/s and the imaging speed is slower than 33 frames/s.

Forming of film surface of very viscous liquid flowing with gas in pipes

The study presents the possible use of optoelectronic system for the measurement of the values, which are specific for hydrodynamics of two-phase gas liquid flow in vertical pipes, where a very-high-viscosity liquid forms a falling film in a pipe. The experimental method was provided, and the findings were presented and analysed for selected values, which characterize the two-phase flow. Attempt was also made to evaluate the effects of flow parameters and properties of the liquid on the gas-liquid interface value, which is decisive for the conditions of heat exchange and mass transfer in falling film equipment. The nature and form of created waves at various velocities were also described.

Uncertainty quantification and global sensitivity analysis of mechanistic one-dimensional models and flow pattern transition boundaries predictions for two-phase pipe flows

The prediction of uncertainties is a growing interest in flow assurance industrial applications, but only few works have been presented on this topic. In this work, an uncertainty quantification and a global sensitivity analysis are performed to quantify the level of confidence in predictions of one-dimensional mechanistic models considering different two-phase flow regimes. A method is proposed for this purpose accounting for the effect of several variables on pressure drop and hold-up predictions by the well-known one-dimensional two-fluid model, such as fluid flow rates, geometry (the inclination angle and the pipe diameter), and fluid properties (density and viscosity); the case of a non-Newtonian shear-thinning fluid behaviour is also considered. Flow pattern transition boundaries, including the stability of the stratified flow regime, are included in this analysis. Monte Carlo simulations were used for the uncertainty quantification while different approaches for the sensitivity analysis (scatter plot, linear regression, the Morris’s method, and the Sobol’s Method) were used and compared to identify the best tool for this family of models. The Sobol’s method appears to be the most convenient approach and a discussion is provided considering different practical cases for gas/liquid and liquid/liquid systems. The most critical input parameters in terms of uncertainty are rigorously identified case by case. A way to reduce the output uncertainty is indicated by the interpretation of the results of the global sensitivity analysis. The conclusions of this analysis gives new insights regarding the degree of uncertainties in predictions of one-dimensional mechanistic models.